The summation in Eq.(4.13) cannot be implemented in practice because
the ``ideal lowpass filter'' impulse response actually extends
from minus infinity to infinity. It is necessary in practice to window the ideal impulse response so as to make it finite. This is the basis
of the window method for digital filter design
[115,362]. While many other filter design techniques
exist, the window method is simple and robust, especially for very
long impulse responses. In the case of the algorithm presented below,
the filter impulse response is very long because it is heavily
oversampled. Another approach is to design optimal decimated
``sub-phases'' of the filter impulse response, which are then
interpolated to provide the ``continuous'' impulse response needed for
the algorithm [358].

If we truncate at the fifth zero-crossing to the left and the
right of the origin, we obtain the frequency response shown in
Fig.4.24. Note that the stopband exhibits only slightly
more than 20 dB rejection.

Figure 4.24:
Frequency response of
the ideal lowpass filter after rectangularly windowing the ideal
(sinc) impulse response at the fifth zero crossing to the left and
right of the time origin. The vertical axis is in units of decibels
(dB), and the horizontal axis is labeled in units of spectral samples
between plus and minus half the sampling rate.

If we instead use the Kaiser window [221,438] to
taper to zero by the fifth zero-crossing to the left and the
right of the origin, we obtain the frequency response shown in
Fig.4.25. Note that now the stopband starts out close to
dB. The Kaiser window has a single parameter which can be used
to modify the stop-band attenuation, trading it against the transition
width from pass-band to stop-band.

Figure 4.25:
Frequency response of the
ideal lowpass filter Kaiser windowed at the fifth zero crossing to the
left and right.